ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Conic Sections (Set 7): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review. Covers 30…
What this test covers
The first 5 questions of this test, with the correct answer and explanation.
Q1. A focal chord of the parabola y² = 12x is inclined at 45° to the axis. Its length is:
Answer: A. 24
Explanation: For y² = 4ax a focal chord making angle θ with the axis has length 4a/sin²θ. Here 4a = 12 and sin²45° = 1/2, so the length is 24.
Q2. The tangent to y² = 12x with slope 1 touches the parabola at the point:
Answer: C. (3, 6)
Explanation: For y² = 4ax the tangent of slope m is y = mx + a/m, so here y = x + 3. Substituting into y² = 12x: (x − 3)² = 0, so the point of contact is x = 3, y = 6.
Q3. The normal to y² = 4x at the point (4, 4) meets the parabola again at the point:
Answer: D. (9, −6)
Explanation: For the point with parameter t = 2, the normal meets the parabola again at t′ = −t − 2/t = −3, i.e. (t′², 2t′) = (9, −6). Check: the normal y + 2x = 12 passes through both (4, 4) and (9, −6).
Q4. For the ellipse x²/25 + y²/9 = 1, the area of the triangle formed by the two foci and an end of the minor axis is:
Answer: C. 12
Explanation: Here c² = 25 − 9 = 16, so the foci are (±4, 0), 8 apart. The minor-axis end (0, 3) is at height 3. Area = ½·8·3 = 12.
Q5. For a point on the ellipse x²/25 + y²/9 = 1 with abscissa 5/2, the product of its distances from the two foci is:
Answer: B. 21
Explanation: The focal distances are a ± ex with e = 4/5: 5 − 2 = 3 and 5 + 2 = 7. Their product is 21.
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Q6. The acute angle between the asymptotes of the hyperbola x² − 3y² = 3 is:
Q7. The distance between the directrices of the hyperbola x²/16 − y²/9 = 1 is:
Q8. The equation of the circle having the segment joining (−1, 2) and (5, −6) as a diameter is:
Q9. The line y = x + 1 cuts the circle x² + y² = 25 in a chord of length:
Q10. The length of the tangent from (7, 1) to the circle x² + y² − 2x − 4y − 4 = 0 is:
Q11. The equation of the tangent to the circle x² + y² − 2x − 4y − 3 = 0 at the point (3, 4) is:
Q12. The circles x² + y² = 16 and x² + y² − 8x = 0 have a common chord of length:
Q13. The circle (x − 3)² + y² = r² touches the circle x² + y² = 4. The sum of all possible values of r is:
Q14. A point moves so that the sum of its distances from (3, 0) and (−3, 0) is always 10. Its locus is:
Q15. The focus of the parabola x² − 4x − 8y + 20 = 0 is:
Q16. The ellipse 4x² + 9y² − 16x + 18y − 11 = 0 has a latus rectum of length:
Q17. The rectangular hyperbola xy = 18 has its two vertices at a distance apart of:
Q18. The conic x² + 4xy + 4y² + 2x − y + 3 = 0 is a:
Q19. The tangents to the ellipse x²/9 + y²/4 = 1 with slope 1 are:
Q20. The line y = 2x + k is tangent to the hyperbola x²/9 − y²/4 = 1 for k equal to:
Q21. An ellipse has its latus rectum equal to half of its major axis. Its eccentricity is:
Q22. A point of the parabola y² = 16x has abscissa 5. Its distance from the focus is:
Q23. The locus of a point whose distance from (2, 0) is half its distance from the line x = 8 is:
Q24. The line y = 2x + k touches the circle x² + y² = 5 for k equal to:
Q25. The angle between the two tangents drawn from the point (0, √2) to the circle x² + y² = 1 is:
Q26. The chord of contact of the tangents drawn from (4, 3) to the circle x² + y² = 9 has the equation:
Q27. The vertex and the two ends of the latus rectum of the parabola y² = 12x form a triangle of area:
Q28. The tangent to the ellipse x²/25 + y²/9 = 1 at the end (4, 9/5) of its latus rectum is:
Q29. A hyperbola with vertices (±3, 0) passes through the point (5, 16/3). Its latus rectum has length:
Q30. The point (2 + 3 cos θ, −1 + 3 sin θ) traces the circle: